In the Admin section, if you navigate to 'Web Services' there is a section title 'API Call Information'. If you click the number after 'Requests in the Last 7 Days:' it'll show you the calls made and by which user.
Make sure to post your questions to Products. A mod just moved your thread today.
In addition to Anna's pointer, there's an API call to get daily usage.
That you are running out of daily calls isn't surprising. Most commercial integrations don't pay correct attention to the quite low daily limit (probably 10,000 for your instance). Custom integrations are more likely to be attentive to this critical area. I recommend passing all API traffic through an accounting gateway such as Apigee.